## Bulk Modulus

We can expand the determinant of the tensor det I + 2E to find det I + 2E 1 + 2Ie + 4IIE + 8IIIE but for small strains, IE IIE IIIE since the first term is linear in E, the second is quadratic, and the third is cubic. Therefore, we can approximate det I + 2E 1 + 2IE, hence we define the volumetric dilatation as this quantity is readily measurable in an experiment.

## Conservation of Energy First Principle of Thermodynamics

30 The first principle of thermodynamics relates the work done on a (closed) system and the heat transfer into the system to the change in energy of the system. We shall assume that the only energy transfers to the system are by mechanical work done on the system by surface traction and body forces, by heat transfer through the boundary. 6.4.1 Spatial Gradient of the Velocity 31 We define L as the spatial gradient of the velocity and in turn this gradient can be decomposed into a symmetric rate...

## Conservation of Mass Continuity Equation Spatial Form

12 If we consider an arbitrary volume V, fixed in space, and bounded by a surface S. If a continuous medium of density p fills the volume at time t, then the total mass in V is where p(x, t) is a continuous function called the mass density. We note that this spatial form in terms of x is most common in fluid mechanics. 13 The rate of increase of the total mass in the volume is 14 The Law of conservation of mass requires that the mass of a specific portion of the continuum remains constant....

## Draft

1 MATHEMATICAL PRELIMINARIES Part I Vectors and Tensors 1-1 1.1 Vectors 1.1.1 Operations 1.1.2 Coordinate 1.1.2.1 jGeneral 1.1.2.1.1 jContravariant 1.1.2.1.2 Covariant 1.1.2.2 Cartesian Coordinate System 1-6 1.2 Tensors 1.2.1 Indicial 1.2.2 Tensor 1.2.2.2 Multiplication by a 1.2.2.4.1 Outer 1.2.2.4.2 Inner 1.2.2.4.3 Scalar 1.2.2.4.4 Tensor Product 1-11 1.2.2.5 Product of Two Second-Order Tensors 1-13 1.2.4 Rotation of 1.2.5 Trace 1.2.6 Inverse Tensor 1.2.7 Principal Values and Directions of...

## Dx

In 2D, this results in (by setting i d 2 31 dx3dx1 d2 11 dx2dx3 d2 22 dx3dx1 d2 33 (4.141-a) (4.141-b) (4.141-c) (4.141-d) (4.141-e) (4.141-f) E 1 ( dxk dxk x Eij 2 dXi dXj ) Ui 2 dXj T 8Xi dXi dXj) U1 E* 1 ( dui 1 duj duk duk or Eij 2 dxj T dxi dxi dx, J ui E* 1 (dui 1 duj A Eij 2 dxj ' dxi J r 1 ( dui + duj + 1 ( dui duj i, X Intrc 2 dXj dXi J 2 dXj OXiJiUXj g (uVx + Vxu) + 1 (uVx - Vxu) -dX 2 dxj T dxi J 2 dxj dxi J j (uVx + Vxu) + 1 (uVx - Vxu) -dx 95 When he compatibility equation is...

## E a oij

Where a is the linear coefficient of thermal expansion. 69 Inserting the preceding two equation into Hooke's law (Eq. 7.51) yields which is known as Duhamel-Neumann relations. 70 If we invert this equation, we obtain the thermoelastic constitutive equation Tij XSijEkk + 2pEij - (3A + 2p)a5ij( - o) 71 Alternatively, if we were to consider the derivation of the Green-elastic hyperelastic equations, (Sect. 7.1.5), we required the constants c to c6 in Eq. 7.22 to be zero in order that the stress...

## Entropy

46 The basic criterion for irreversibility is given by the second principle of thermodynamics through the statement on the limitation of entropy production. This law postulates the existence of two distinct state functions 0 the absolute temperature and S the entropy with the following properties 2. Entropy is an extensive property, i.e. the total entropy is in a system is the sum of the entropies of its parts. where ds(e) is the increase due to interaction with the exterior, and ds(i) is the...

## Equations of Conditions

25 If a structure has an internal hinge (which may connect two or more substructures), then this will provide an additional equation (EM 0 at the hinge) which can be exploited to determine the reactions. 26 Those equations are often exploited in trusses (where each connection is a hinge) to determine reactions. 27 In an inclined roller support with Sx and Sy horizontal and vertical projection, then the reaction R would have, Fig. 12.2. Figure 12.2 Inclined Roller Support 12.2.4 Static...

## Example

A 20 ft long, uniformly loaded, beam is simply supported at one end, and rigidly connected at the other. The beam is composed of a steel tube with thickness t 0.25 in. Select the radius such that Vmax < 18 ksi, and Amax < L 360. 1. Steel has E 29, 000 ksi, and from above Mmax Wf-, Amax wEi , and I nr3t. 3. We next seek a relation between maximum deflection and radius (1) k ft(20)4 ft4(12)3 in3 ft3 (185)(29,000) ksi(3.14)r3(0.25) in 5. We now set those two values equal to their respective...

## F Elements of Heat Transfer

63 One of the relations which we will need is the one which relates temperature to heat flux. This constitutive realtion will be discussed in the next chapter under Fourrier's law. 64 However to place the reader in the right frame of reference to understand Fourrier's law, this section will provide some elementary concepts of heat transfer. 65 There are three fundamental modes of heat transfer Conduction takes place when a temperature gradient exists within a material and is governed by...

## F Thermodynamic Approach State Variables

1 The method of local state postulates that the thermodynamic state of a continuum at a given point and instant is completely defined by several state variables (also known as thermodynamic or independent variables). A change in time of those state variables constitutes a thermodynamic process. Usually state variables are not all independent, and functional relationships exist among them through equations of state. Any state variable which may be expressed as a single valued function of a set...

## Gibbs Relation

8 From the chain rule we can express substituting into Clausius-Duhem inequality of Eq. 6.66 but the second principle must be satisfied for all possible evolution and in particular the one for which D 0, dVp 0 and V9 0 for any value of dt thus the coefficient of dt is zero or and if we adopt the differential notation, we obtain Gibbs relation For fluid, the Gibbs relation takes the form where p is the thermodynamic pressure and the thermodynamic tension conjugate to the specific volume v is p,...

## Gradient Scalar

16 The gradient of a scalar field g(x) is a vector field Vg(x) such that for any unit vector v, the directional derivative dg ds in the direction of v is given by < < Calculus'VectorAnalysis1 V (xA2z, -2yA3zA2, xy 2z Div V, Cartesian x, y, z 6 z2 y2 + xy2 +2 xz < < Graphics,PlotField3D1 PlotVectorField3D (x*2 z, -2yA3zA2, xy 2 z , (x, -10, 10 , (y, -10, 10 , (z, -10, 10 , Axes-> Automatic, AxesLabel -> (X, Y, Z Div Curl V, Cartesian x, y, z , Cartesian x, y, z 0 Figure 3.8...

## Ideal Strength in Terms of Physical Parameters

9 We shall first derive an expression for the ideal strength in terms of physical parameters, and in the next section the strength will be expressed in terms of engineering ones. Solution I Force being the derivative of energy, we have F , thus F 0 at a a0, Fig. 11.4, and is maximum at the inflection point of the U0 a curve. Hence, the slope of the force displacement curve is the stiffness of the atomic spring and should be related to E. If we let x a a0, then the strain would be equal to e ....

## Intermezzo

In light of the lengthy and rigorous derivation of the fundamental equations of Continuum Mechanics in the preceding chapter, the reader may be at a loss as to what are the most important ones to remember. Hence, since the complexity of some of the derivation may have eclipsed the final results, this handout seeks to summarize the most fundamental relations which you should always remember. Engineering Strain Equilibrium Boundary Conditions Energy Potential Hooke's Law (8.1-d) (8.1-e) (8.1-f)...

## Internal Strain Energy

Ii The strain energy density of an arbitrary material is defined as, Fig. 13.1 The complementary strain energy density is defined The strain energy itself is equal to 14 To obtain a general form of the internal strain energy, we first define a stress-strain relationship accounting for both initial strains and stresses where D is the constitutive matrix (Hooke's Law) e is the strain vector due to the displacements u e0 is the initial strain vector a0 is the initial stress vector and a is the...

## Invariants

33 The principal stresses are physical quantities, whose values do not depend on the coordinate system in which the components of the stress were initially given. They are therefore invariants of the stress state. 34 When the determinant in the characteristic Eq. 2.32 is expanded, the cubic equation takes the form where the symbols Ia, IIa and IIIa denote the following scalar expressions in the stress components 35 In terms of the principal stresses, those invariants can be simplified into...

## K a

Where p 1 tr (a) is the pressure, and a' a pI is the stress deviator. 7.3.5.1.1.3 Restriction Imposed on the Isotropic Elastic Moduli but since dW is a scalar invariant (energy), it can be expressed in terms of volumetric (hydrostatic) and deviatoric components as substituting p -Ke and oj 2GEj, and integrating, we obtain the following expression for the isotropic strain energy W 1 Ke2 + GEj Ej and since positive work is required to cause any deformation W > 0 thus ruling out K G 0, we are...

## Na An

In indicial notation this can be rewritten as in matrix notation this corresponds to where I corresponds to the identity matrix. We really have here a set of three homogeneous algebraic equations for the direction cosines n. 27 Since the direction cosines must also satisfy they can not all be zero. hence Eq.2.28 has solutions which are not zero if and only if the determinant of the coefficients is equal to zero, i.e 28 For a given set of the nine stress components, the preceding equation...

## Potential Energy

29 The potential of external work W in an arbitrary system is defined as where u are the displacements, b is the body force vector t is the applied surface traction vector r is that portion of the boundary where t is applied, and P are the applied nodal forces. 30 Note that the potential of the external work (W) is different from the external work itself (W) 31 The potential energy of a system is defined as 32 Note that in the potential the full load is always acting, and through the...

## Preliminary Definitions

6 Work is defined as the product of a force and displacement 7 Energy is a quantity representing the ability or capacity to perform work. 8 The change in energy is proportional to the amount of work performed. Since only the change of energy is involved, any datum can be used as a basis for measure of energy. Hence energy is neither created nor consumed. 9 The first principle of thermodynamics (Eq. 6.44), states The time-rate of change of the total energy (i.e., sum of the kinetic energy and...

## Products Outer Product

41 The outer product of two tensors (not necessarily of the same type or order) is a set of tensor components obtained simply by writing the components of the two tensors beside each other with no repeated indices (that is by multiplying each component of one of the tensors by every component of the other). For example

## Semi Inverse Method

3 Often a solution to an elasticity problem may be obtained without seeking simulateneous solutions to the equations of motion, Hooke's Law and boundary conditions. One may attempt to seek solutions by making certain assumptions or guesses about the components of strain stress or displacement while leaving enough freedom in these assumptions so that the equations of elasticity be satisfied. 4 If the assumptions allow us to satisfy the elasticity equations, then by the uniqueness theorem, we...

## Shear Moment Diagrams Design Sign Conventions

36 Before we derive the Shear-Moment relations, let us arbitrarily define a sign convention. 37 The sign convention adopted here, is the one commonly used for design purposes4. With reference to Fig. 12.5 Figure 12.5 Shear and Moment Sign Conventions for Design Load Positive along the beam's local y axis (assuming a right hand side convention), that is positive upward. Flexure A positive moment is one which causes tension in the lower fibers, and compression in the upper ones. For frame...

## Simple D Derivation

66 If we consider a unit thickness, 2D differential body of dimensions dx by dy, Fig. 6.4 then 1. Rate of heat generation sink is 2. Heat flux across the boundary of the element is shown in Fig. (note similarity with equilibrium equation) Figure 6.4 Flux Through Sides of Differential Element Figure 6.5 *Flow through a surface r where we define the specific heat c as the amount of heat required to raise a unit mass by one degree. 67 From the first law of thermodaynamics, energy produced I2 plus...

## Simplified Theories Stress Resultants

48 For many applications of continuum mechanics the problem of determining the three-dimensional stress distribution is too difficult to solve. However, in many (civil mechanical)applications, one or more dimensions is are small compared to the others and possess certain symmetries of geometrical shape and load distribution. 49 In those cases, we may apply engineering theories for shells, plates or beams. In those problems, instead of solving for the stress components throughout the body, we...

## Stress Strain Relations

29 In orthogonal curvilinear coordinates, the physical components of a tensor at a point are merely the Cartesian components in a local coordinate system at the point with its axes tangent to the coordinate curves. Hence, For Plane strain problems, from Eq. 7.75 and zz Yrz YOz Trz Tgz 0. 31 Inverting, 32 For plane stress problems, from Eq. 7.78-a and Trz TOz & zz Yrz YOz 0 33 Inverting

## Stress Strain Relations in Generalized Elasticity Anisotropic

28 From Eq. 7.22 and 7.23 we obtain the stress-strain relation for homogeneous anisotropic material which is Hooke's law for small strain in linear elasticity. 29 We also observe that for symmetric cij we retrieve Clapeyron formula 30 In general the elastic moduli cij relating the cartesian components of stress and strain depend on the orientation of the coordinate system with respect to the body. If the form of elastic potential function W and the values cij are independent of the orientation,...

## Symmetry of Stress Tensor

16 From Fig. 2.1 the resultant force exerted on the positive X face is L a 11AX2AX3 a 12AX2 AX3 a 13AX2 AX3 J similarly the resultant forces acting on the positive X2 face are 17 We now consider moment equilibrium (M Fxd). The stress is homogeneous, and the normal force on the opposite side is equal opposite and colinear. The moment (AX2 2)a31 AX1AX2 is likewise balanced by the moment of an equal component in the opposite face. Finally similar argument holds for is The net moment about the X3...

## Theor X

Thus this would be the ideal theoretical strength of steel. 11.2.2 Ideal Strength in Terms of Engineering Parameter 10 We note that the force to separate two atoms drops to zero when the distance between them is a0 + a where a0 corresponds to the origin and a to . Thus, if we take a or A 2a, combined with Eq. 11.11 would yield 11 Alternatively combining Eq. 11.9 with A 2a gives Combining those two equations will give 12 However, since as a first order approximation a a0 then the surface energy...

## Uniqueness of the Elastostatic Stress and Strain Field

15 Because the equations of linear elasticity are linear equations, the principles of superposition may be used to obtain additional solutions from those established. Hence, given two sets of solution T(1, ui1 , and Tj ui2), then Tij Tj Tj and ui u 2) ui1 with bi b 2) bi1 0 must also be a solution. 16 Hence for this difference solution, Eq. 9.18 would yield tiuidT 2 u*di but the left hand side is zero because ti t ti1 0 on T , and ui u 2 nip 0 on Tj, thus u*dl 0. 17 But u* is positive-definite...

## Vectors

2 A vector is a directed line segment which can denote a variety of quantities, such as position of point with respect to another (position vector), a force, or a traction. 3 A vector may be defined with respect to a particular coordinate system by specifying the components of the vector in that system. The choice of the coordinate system is arbitrary, but some are more suitable than others (axes corresponding to the major direction of the object being analyzed). 4 The rectangular Cartesian...

## Virtual Work

23 We define the virtual work done by the load on a body during a small, admissible (continuous and satisfying the boundary conditions) change in displacements. where all the terms have been previously defined and b is the body force vector. 24 Note that the virtual quantity (displacement or force) is one that we will approximate guess as long as it meets some admissibility requirements. 25 Next we shall derive a displacement based expression of 5U for each type of one dimensional structural...

## Akc

Figure 10.1 Torsion of a Circular Bar Figure 10.1 Torsion of a Circular Bar The non zero stress components are obtained from Hooke's law 10 We need to check that this state of stress satisfies equilibrium dTij dxj identically satisfied, whereas the other two yield Physically, this means that equilibrium is only satisfied if the increment in angular rotation (twist per unit length) is a constant. 11 We next determine the corresponding surface tractions. On the lateral surface we have a unit...

## Dx from to

Reactions are determined from the equilibrium equations (+ )ZFX 0 (+ ) SMa 0 (+ t ) EFy 0 (11)(4) + (8)(1O) + (4)(2)(14 + 2) - RPy (18) O RFy 14 k RAy - 11 - 8 - (4)(2) + 14 O RAy 13 k 1. At A the shear is equal to the reaction and is positive. 2. At B the shear drops (negative load) by 11 k to 2 k. 3. At C it drops again by 8 k to 6 k. 4. It stays constant up to D and then it decreases (constant negative slope since the load is uniform and negative) by 2 k per linear foot up to 14 k. 5. As a...

## Equation of State Second Principle of Thermodynamics

43 The complete characterization of a thermodynamic system is said to describe the state of a system (here a continuum). This description is specified, in general, by several thermodynamic and kinematic state variables. A change in time of those state variables constitutes a thermodynamic process. Usually state variables are not all independent, and functional relationships exist among them through equations of state. Any state variable which may be expressed as a single valued function of a...

## Balance of Equations and Unknowns

58 In the preceding sections several equations and unknowns were introduced. Let us count them. for both the coupled and uncoupled cases. 59 Assuming that the body forces bi and distributed heat sources r are prescribed, then we have the following unknowns and in addition the Clausius-Duhem inequality js gt - div q which governs entropy production must hold. P 60 We thus need an additional 16 5 11 additional equations to make the system determinate. These will be later on supplied by Total...

## Potential Energy Derivation

47 From section , if U0 is a potential function, we take its differential 49 We now define the variation of the strain energy density at a point1 Applying the principle of virtual work, Eq. 13.37, it can be shown that 1 Note that the variation of strain energy density is, Uo amp ij, and the variation of the strain energy itself is amp U fn U0dQ. Figure 13.4 Single DOF Example for Potential Energy 51 We have thus derived the principle of stationary value of the potential energy Of all...

## M Example Tangent to a Curve

Determine the unit vector tangent to the curve x t2 1, y 4t 3, z 2t2 6t for t 2. Solution dt t2 1 i 4t 3 j 2t2 6t k 2ti 4j 4t 6 k 2ti 4j 4t 6 k J 2t 2 4 2 4t 6 2 , - i i k for t 2 Mathematica solution is shown in Fig. 3.4 ParametricPlot3D t 2 1, 4 t - 3, 2 t 2 - 6 t , t, 0, 4 ParametricPlot3D t 2 1, 4 t - 3, 2 t 2 - 6 t , t, 0, 4 Figure 3.4 Mathematica Solution for the Tangent to a Curve in 3D

## Force Traction and Stress Vectors

1 There are two kinds of forces in continuum mechanics body forces act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. dF pbdVol. surface forces are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area. 2 The surface force per unit area acting on an element dS is called traction or more accurately stress vector. I tdS if txdS j tydS k tzdS 2.1 Most authors limit the term traction to an...

## Mathematica Assignment and Solution

Connect to Mathematica using the following procedure 5. On the newly opened shell, enter your password first, and then type setenv DISPLAY xxx 0.0 where xxx is the workstation name which should appear on a small label on the workstation itself. and then solve the following problems 1. The state of stress through a continuum is given with respect to the cartesian axes 0x1x2x3 by Determine the stress vector at point P 1,1, a 3 of the plane that is tangent to the cylindrical surface x2 X3 4 at P....

## Load Shear Moment Relations

38 Let us derive the basic relations between load, shear and moment. Considering an infinitesimal length dx of a beam subjected to a positive load5 w x , Fig. 12.6. The infinitesimal section must also be in equilibrium. 39 There are no axial forces, thus we only have two equations of equilibrium to satisfy XFy 0 and XMz 0. 40 Since dx is infinitesimally small, the small variation in load along it can be neglected, therefore we assume w x to be constant along dx. 41 To denote that a small change...

## Principle of Virtual Work and Complementary Virtual Work

33 The principles of Virtual Work and Complementary Virtual Work relate force systems which satisfy the requirements of equilibrium,, and deformation systems which satisfy the requirement of compatibility. 1. In any application the force system could either be the actual set of external loads dp or some virtual force system which happens to satisfy the condition of equilibrium p. This set of external forces will induce internal actual forces da or internal hypothetical forces Sa compatible with...

## Geometric Instability

33 The stability of a structure is determined not only by the number of reactions but also by their arrangement. 34 Geometric instability will occur if 1. All reactions are parallel and a non-parallel load is applied to the structure. 2. All reactions are concurrent, Fig. 12.4. Figure 12.4 Geometric Instability Caused by Concurrent Reactions Figure 12.4 Geometric Instability Caused by Concurrent Reactions 3. The number of reactions is smaller than the number of equations of equilibrium, that is...

## Transversely Isotropic Material

38 A material is transversely isotropic if there is a preferential direction normal to all but one of the three axes. If this axis is x3, then rotation about it will require that cos 6 sin 6 0 sin 6 cos 6 0 0 0 1 substituting Eq. 7.33 into Eq. 7.41, using the above transformation matrix, we obtain C1111 cos4 6 c1111 cos2 6 sin2 6 2cm2 4cm2 sin4 6 c2222 C1122 cos2 6 sin2 6 cm1 cos4 6 cm2 4 cos2 6 sin2 6 cm2 sin4 6 c2211 sin2 6 cos2 6 c2222 cos2 6 c1133 sin2 6 c2233 sin4 6 cm1 cos2 6 sin2 6 2cm2...

## Example Mohrs Circle in Plane Stress

An element in plane stress is subjected to stresses axx 15, ayy 5 and Txy 4. Using the Mohr's circle determine a the stresses acting on an element rotated through an angle 0 40 counterclockwise b the principal stresses and c the maximum shear stresses. Show all results on sketches of properly oriented elements. Solution 1. The center of the circle is located at 2 axx ayy 1 15 5 10. 2.63 2. The radius and the angle 2p are given by R y1 15 5 2 42 6.403 2.64-a 2 4 tan2 w 0.8 38.66 f3 19.33 2.64-b...

## Summary

60 Summary of Virtual work methods, Table 13.2. Displacement strains Forces Stresses KAD Kinematically Admissible Dispacements SAS Statically Admissible Stresses Table 13.2 Comparison of Virtual Work and Complementary Virtual Work 61 A summary of the various methods introduced in this chapter is shown in Fig. 13.7. 62 The duality between the two variational principles is highlighted by Fig. 13.8, where beginning with kinematically admissible displacements, the principle of virtual work provides...

## Boundary Conditions

4 In describing the boundary conditions B.C. , we must note that 1. Either we know the displacement but not the traction, or we know the traction and not the corresponding displacement. We can never know both a priori. 2. Not all boundary conditions specifications are acceptable. For example we can not apply tractions to the entire surface of the body. Unless those tractions are specially prescribed, they may not necessarily satisfy equilibrium. 5 Properly specified boundary conditions result...

## Cauchys Reciprocal Theorem

21 If we consider t1 as the traction vector on a plane with normal n1, and t2 the stress vector at the same point on a plane with normal n2, then t1 n1 and t2 n2a ti L 1JM and 2 L -2JH If we postmultiply the first equation by n2 and the second one by n1, by virtue of the symmetry of 0 we have Figure 2.4 Cauchy's Reciprocal Theorem Figure 2.4 Cauchy's Reciprocal Theorem 22 In the special case of two opposite faces, this reduces to 23 We should note that this theorem is analogous to Newton's...

## Example Principal Stresses

The stress tensor is given at a point by determine the principal stress values and the corresponding directions. Solution Or upon expansion and simplification A 2 A 4 A 1 0, thus the roots are a l 4, a 2 1 and a 3 2. We also note that those are the three eigenvalues of the stress tensor. If we let xl axis be the one corresponding to the direction of a 3 and n3 be the direction cosines of this axis, then from Eq. 2.28 we have Similarly If we let x2 axis be the one corresponding to the direction...

## Size Effect Griffith Theory

13 In his quest for an explanation of the size effect, Griffith came across Inglis's paper, and his strike of genius was to assume that strength is reduced due to the presence of internal flaws. Griffith postulated that the theoretical strength can only be reached at the point of highest stress concentration, and accordingly the far-field applied stress will be much smaller. 14 Hence, assuming an elliptical imperfection, and from equation 11.2 a is the stress at the tip of the ellipse which is...

## Basic Kinematic Assumption Curvature

47 Fig.12.7 shows portion of an originally straight beam which has been bent to the radius p by end couples M. support conditions, Fig. 12.1. It is assumed that plane cross-sections normal to the Figure 12.7 Deformation of a Beam under Pure Bending length of the unbent beam remain plane after the beam is bent. Figure 12.7 Deformation of a Beam under Pure Bending length of the unbent beam remain plane after the beam is bent. 48 Except for the neutral surface all other longitudinal fibers either...

## Isotropic Material

40 An isotropic material is symmetric with respect to every plane and every axis, that is the elastic properties are identical in all directions. 41 To mathematically characterize an isotropic material, we require coordinate transformation with rotation about x2 and xi axes in addition to all previous coordinate transformations. This process will enforce symmetry about all planes and all axes. 42 The rotation about the x2 axis is obtained through cos 6 0 sin 6 0 1 0 sin 6 0 cos 6 we follow a...

## Example Stress Vector normal to the Tangent of a Cylinder

The stress tensor throughout a continuum is given with respect to Cartesian axes as Determine the stress vector or traction at the point P 2, VS of the plane that is tangent to the cylindrical surface x2 x2 4 at P, Fig. 3.9. Figure 3.9 Radial Stress vector in a Cylinder Figure 3.9 Radial Stress vector in a Cylinder At point P, the stress tensor is given by The unit normal to the surface at P is given from Thus the traction vector will be determined from 20 We can also define the gradient of a...

## Operations

Addition of two vectors a b is geometrically achieved by connecting the tail of the vector b with the head of a, Fig. 1.2. Analytically the sum vector will have components a b a2 b2 a3 b3 J. Scalar multiplication aa will scale the vector into a new one with components aai aa2 aa3 J. Vector Multiplications of a and b comes in three varieties Dot Product or scalar product is a scalar quantity which relates not only to the lengths of the vector, but also to the angle between them. a-b a UK b cos9...

## V

Finally, the displacement components can be obtained by integrating the above equations 10.2.2.3 Example Thick-Walled Cylinder 4i If we consider a circular cylinder with internal and external radii a and b respectively, subjected to internal and external pressures p and po respectively, Fig. 10.2, then the boundary conditions for the plane strain problem are Ttt Pi at r Oi Ttt -po at r b 42 These Boundary conditions can be easily shown to be satisfied by the following stress field These...

## Rotation of Axes

55 The rule for changing second order tensor components under rotation of axes goes as follow ajTjqvq From Eq. 1.39-a 1.71 But we also have Ui Tipvp again from Eq. 1.39-a in the barred system, equating these two expressions we obtain By extension, higher order tensors can be similarly transformed from one coordinate system to another. 56 If we consider the 2D case, From Eq. 1.38 2 sin 2aTxx sin 2aTyy 2 cos 2aTxy 2 sin 2aTxx sin 2aTyy 2 cos 2aTxy sin2 aTxx cos a cos aTyy 2 sin aTxy 0...

## J Contravariant Transformation

12 The vector representation in both systems must be the same V Vqbq Vkbk Vk bkbq v - Vk b bq 0 1.23 since the base vectors bq are linearly independent, the coefficients of bq must all be zero hence showing that the forward change from components vk to vq used the coefficients bqk of the backward change from base bq to the original bk. This is why these components are called contravariant. 13 Generalizing, a Contravariant Tensor of order one recognized by the use of the superscript transforms a...

## Rayleigh Ritz Method

55 Continuous systems have infinite number of degrees of freedom, those are the displacements at every point within the structure. Their behavior can be described by the Euler Equation, or the partial differential equation of equilibrium. However, only the simplest problems have an exact solution which satisfies equilibrium, and the boundary conditions . 56 An approximate method of solution is the Rayleigh-Ritz method which is based on the principle of virtual displacements. In this method we...

## Arch Elasticity

51 Fig. 2.10 illustrates the stresses acting on a differential element of a shell structure. The resulting forces in turn are shown in Fig. 2.11 and for simplification those acting per unit length of the middle surface are shown in Fig. 2.12. The net resultant forces are given by Figure 2.9 Mohr Circle for Stress in 3D Figure 2.10 Differential Shell Element, Stresses Figure 2.10 Differential Shell Element, Stresses Figure 2.11 Differential Shell Element, Forces Figure 2.11 Differential Shell...

## Spherical and Deviatoric Stress Tensors

36 If we let o denote the mean normal stress p 0 -p 3 011 022 033 3on tr a then the stress tensor can be written as the sum of two tensors Hydrostatic stress in which each normal stress is equal to p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. Deviatoric Stress which causes the change in shape.

## Continuum Mechanics

Saouma Exam I Closed notes , March 27, 1998 3 Hours There are 19 problems worth a total of 63 points. Select any problems you want as long as the total number of corresponding points is equal to or larger than 50. 1. 2 pts Write in matrix form the following 3rd order tensor Dijk in R2 space. i,j,k range from 1 to 2. 2. 2 pts Solve for Eijai in indicial notation. 3. 4 pts if the stress tensor at point P is given by determine the traction or stress vector t on the plane passing...

## Elastic Potential or Strain Energy Function

16 Green defined an elastic material as one for which a strain-energy function exists. Such a material is called Green-elastic or hyperelastic if there exists an elastic potential function W or strain energy function, a scalar function of one of the strain or deformation tensors, whose derivative with respect to a strain component determines the corresponding stress component. 17 For the fully recoverable case of isothermal deformation with reversible heat conduction we have hence W p0 is an...

## Mohrs Circle for Plane Stress Conditions

41 The Mohr circle will provide a graphical mean to contain the transformed state of stress axx,ayy, xy at an arbitrary plane inclined by a in terms of the original one axx, ayy, axy . cos2 a 1 c s2a sin2 a -c s2 cos 2a cos2 a - sin2 a sin 2a 2 sin a cos a into Eq. 2.49 and after some algebraic manipulation we obtain 7 axx Vyy T. Jxx - yy cos2a aXy sin2a 2.57-a TXy axy cos 2a - ctxx - yy sin2a 2.57-b 43 Points axx,axy , axx, 0 , ayy, 0 and axx ayy 2, 0 are plotted in the stress representation...

## Navier Cauchy Equations

11 One such approach is to substitute the displacement-strain relation into Hooke's law resulting in stresses in terms of the gradient of the displacement , and the resulting equation into the equation of motion to obtain three second-order partial differential equations for the three displacement components known as Navier's Equation Figure 9.3 Fundamental Equations in Solid Mechanics

## Cartesian Coordinate System

17 If we consider two different sets of cartesian orthonormal coordinate systems ei, e2, e3 and e, e2, e3 , any vector v can be expressed in one system or the other is To determine the relationship between the two sets of components, we consider the dot product of v with one any of the base vectors 19 We can thus define the nine scalar values which arise from the dot products of base vectors as the direction cosines. Since we have an orthonormal system, those values are nothing else than the...

## Shear Moment and Deflection Diagrams for BEAMS

Adapted from 1 Simple Beam uniform Load 2 Simple Beam Unsymmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 4 Simple Beam Uniform Load Partially Distributed 4 Simple Beam Uniform Load Partially Distributed 5 Simple Beam Concentrated Load at Center at x 2 when x lt 2 whenx lt 2 at x L 6 Simple Beam Concentrated Load at Any Point

## Strain Decomposition

72 In this section we first seek to express the relative displacement vector as the sum of the linear Lagrangian or Eulerian strain tensor and the linear Lagrangian or Eulerian rotation tensor. This is restricted to small strains. 73 For finite strains, the former additive decomposition is no longer valid, instead we shall consider the strain tensor as a product of a rotation tensor and a stretch tensor. 4.3.1 jLinear Strain and Rotation Tensors 74 Strain components are quantitative measures of...

## Euler Equation

1 The fundamental problem of the calculus of variation1 is to find a function u x such that 2 We define u x to be a function of x in the interval a, b , and F to be a known function such as the energy density . 3 We define the domain of a functional as the collection of admissible functions belonging to a class of functions in function space rather than a region in coordinate space as is the case for a function . 4 We seek the function u x which extremizes n. 5 Letting u to be a family of...

## Principle of Virtual Work

35 Derivation of the principle of virtual work starts with the assumption of that forces are in equilibrium and satisfaction of the static boundary conditions. The Equation of equilibrium Eq. 6.26 which is rewritten as where b representing the body force. In matrix form, this can be rewritten as Note that this equation can be generalized to 3D. 37 The surface r of the solid can be decomposed into two parts r and T where tractions and displacements are respectively specified. t t on rt Natural...

## Saint Venants Principle

Is This famous principle of Saint Venant was enunciated in 1855 and is of great importance in applied elasticity where it is often invoked to justify certain simplified solutions to complex problem. In elastostatics, if the boundary tractions on a part r of the boundary r are replaced by a statically equivalent traction distribution, the effects on the stress distribution in the body are negligible at points whose distance from r is large compared to the maximum distance between points of ri....

## Principal Strains Strain Invariants Mohr Circle

So Determination of the principal strains E 3 lt E 2 lt E i , strain invariants and the Mohr circle for strain parallel the one for stresses Sect. 2.4 and will not be repeated here. where the symbols IE, IIE and IIIE denote the following scalar expressions in the strain components Ie En E22 E33 En tr E 4.164 IIe E11E22 E22E33 E33E11 E223 E31 E222 4.165 2 EjEij EiiEjj 2Eij Eij 2IE 4.166 IIIe detE eijk epqr EipEjqEkr 4.168 87 In terms of the principal strains, those invariants can be simplified...

## Statics Equilibrium

8 Any structural element, or part of it, must satisfy equilibrium. 1 So far we have restricted ourselves to a continuum, in this chapter we will consider a structural element. Summation of forces and moments, in a static system must be equal to zero2. In a 3D cartesian coordinate system there are a total of 6 independent equations of equilibrium SFx SFy SF 0 SMx SMy SM2 0 11 In a 2D cartesian coordinate system there are a total of 3 independent equations of equilibrium 12 All the externally...

## Principle of Complementary Virtual Work

40 Derivation of the principle of complementary virtual work starts from the assumption of a kinematicaly admissible displacements and satisfaction of the essential boundary conditions. 41 Whereas we have previously used the vector notation for the principle of virtual work, we will now use the tensor notation for this derivation. 42 The kinematic condition strain-displacement 43 The essential boundary conditions are expressed as 44 The principle of virtual complementary work or more...

## Coordinate Transformation jGeneral Tensors

9 Let us consider two bases bj x x3 and bj xi, x2x3 , Fig. 1.5. Each unit vector in one basis must be a linear combination of the vectors of the other basis bj- apbp and bk b bg 1.20 summed on p and q respectively where apP subscript new, superscript old and bk are the coefficients for the forward and backward changes respectively from b to b respectively. Explicitly Figure 1.5 Coordinate Transformation Figure 1.5 Coordinate Transformation 10 The transformation must have the determinant of its...

## Introduction

1 In Eq. we showed that around a circular hole in an infinite plate under uniform traction, we do have a stress concentration factor of 3. 2 Following a similar approach though with curvilinear coordinates , it can be shown that if we have an elliptical hole, Fig. , we would have We observe that for a b, we recover the stress concentration factor of 3 of a circular hole, and that for a degenerated ellipse, i.e a crack there is an infinite stress. Alternatively, the stress can be expressed in...

## Derivative WRT to a Scalar

4 The derivative of a vector p u with respect to a scalar u, Fig. 3.2 is defined by dp lim p u Au p u du Au o Au 5 If p u is a position vector p u x u i y u j z u k, then ContourPlot Exp - x'2 y 2 , x, -2, 2 , y, -2, 2 , ContourShading - gt False Plot3D Exp - xA2 y 2 , x, -2, 2 , y, -2, 2 , FaceGrids - gt All Figure 3.1 Examples of a Scalar and Vector Fields Figure 3.1 Examples of a Scalar and Vector Fields Figure 3.2 Differentiation of position vector p is a vector along the tangent to the...

## Lagrangian Stresses Piola Kirchoff Stress Tensors

96 In Sect. 2.2 the discussion of stress applied to the deformed configuration dA using spatial coordiantes x , that is the one where equilibrium must hold. The deformed configuration being the natural one in which to characterize stress. Hence we had note the use of T instead of a . Hence the Cauchy stress tensor was really defined in the Eulerian space. 97 However, there are certain advantages in referring all quantities back to the undeformed configuration Lagrangian of the body because...

## Gauss Divergence Theorem

6 The divergence theorem also known as Ostrogradski's Theorem comes repeatedly in solid mechanics and can be stated as follows That is the integral of the outer normal component of a vector over a closed surface which is the volume flux is equal to the integral of the divergence of the vector over the volume bounded by the closed surface. 7 For 2D-1D transformations, we have 8 This theorem is sometime refered to as Green's theorem in space.

## Strain Tensor

8 Following the simplified and restrictive introduction to strain, we now turn our attention to a rigorous presentation of this important deformation tensor. 9 The approach we will take in this section is as follows 1. Define Material fixed, Xj and Spatial moving, Xj coordinate systems. 2. Introduce the notion of a position and of a displacement vector, U, u, with respect to either coordinate system . 3. Introduce Lagrangian and Eulerian descriptions. 4. Introduce the notion of a material...

## Cartesian Coordinates Plane Strain

16 If the deformation of a cylindrical body is such that there is no axial components of the displacement and that the other components do not depend on the axial coordinate, then the body is said to be in a state of plane strain. If e3 is the direction corresponding to the cylindrical axis, then we have Ui Ui xi,x2 , U2 U2 xi,X2 , U3 0 and the strain components corresponding to those displacements are and the non-zero stress components are Tii, Ti2, T22, T33 where 17 Considering a static...

## F Experimental Measurement of Strain

90 Typically, the transducer to measure strains in a material is the strain gage. The most common type of strain gage used today for stress analysis is the bonded resistance strain gage shown in Figure 4.9. Figure 4.9 Bonded Resistance Strain Gage 91 These gages use a grid of fine wire or a metal foil grid encapsulated in a thin resin backing. The gage is glued to the carefully prepared test specimen by a thin layer of epoxy. The epoxy acts as the carrier matrix to transfer the strain in the...

## Hydrostatic and Deviatoric Strain

85 The lagrangian and Eulerian linear strain tensors can each be split into spherical and deviator tensor as was the case for the stresses. Hence, if we define then the components of the strain deviator E' are given by We note that E' measures the change in shape of an element, while the spherical or hydrostatic strain iel represents the volume change. es disusing lt C Jo h0 we t,ai t tOlst MatrixForm Tfirst . 0, 1, 0 gt We note that this vector is in the same direction as t e Cauchy stress...